7.7: Extension : Self-Similarity

Learning Objectives

Self-Similar: When one part of an object can be enlarged (or shrunk) to look like the whole object.

To explore self-similarity, we will go through some examples. Typically, each step of repetition is called an iteration. The first level is called Stage 0.

Sierpinski Triangle

The Sierpinski triangle iterates a triangle by connecting the midpoints of the sides and shading the central triangle (Stage 1). Repeat this process for the unshaded triangles in Stage 1 to get Stage 2.

Example 1: Determine the number of shaded and unshaded triangles in each stage of the Sierpinkski triangles. Determine if there is a pattern.

Solution:

Stage 0

Stage 1

Stage 2

Stage 3

Unshaded

1

3

9

27

Shaded

0

1

4

13

The unshaded triangles seem to be powers of The shaded triangles are add the previous number of unshaded triangles to the total. For Example, Stage 4 would equal 9 + 13 shaded triangles.

Fractals

A fractal is another self-similar object that is repeated at smaller scales. Below are the first three stages of the Koch snowflake.

Example 2: Determine the number of edges and the perimeter of each snowflake.

Stage 0

Stage 1

Stage 2

Number of Edges

3

12

48

Edge Length

1

Perimeter

3

4

The Cantor Set

The Cantor set is another fractal that consists of dividing a segment into thirds and then erasing the middle third.

Review Questions

Draw Stage 4 of the Cantor set.

Use the Cantor Set to fill in the table below.

Number of Segments

Length of each Segment

Total Length of the Segments

Stage 0

1

1

1

Stage 1

2

Stage 2

4

Stage 3

Stage 4

Stage 5

How many segments are in Stage ?

Draw Stage 3 of the Koch snowflake.

A variation on the Sierpinski triangle is the Sierpinski carpet, which splits a square into 9 equal squares, coloring the middle one only. Then, split the uncolored squares to get the next stage. Draw the first 3 stages of this fractal.

How many colored vs. uncolored square are in each stage?

Fractals are very common in nature. For example, a fern leaf is a fractal. As the leaves get closer to the end, they get smaller and smaller. Find three other examples of fractals in nature.